The objective of this paper is to give an overview of some recently developed sum rules and physical bounds in scattering and antenna theory. The sum rules are based on integral identities for Herglotz functions that relate the quantity of interest with its low and high-frequency behavior. The sum rules are transformed to bounds by estimating the integrals and applying variational results to the parameters that appear in the asymptotic expansions. The theoretical findings are exemplified by numerical results for various scattering and antenna configurations.

The objective of this paper is to review some recently developed sum rules and physical bounds in scattering and antenna theory. The sum rules are based on identities for Herglotz functions that relate the quantity of interest integrated over all wavelengths with its static polarizability dyadics. They are transformed to physical bounds by applying variational principles for the polarizability dyadics together with various estimates of the integrals. The theoretical findings are exemplified by numerical results for several configurations.

An antenna identity, derived from the forward scattering sum rule, shows that the partial realized gain of an antenna is related to the polarizability of the antenna structure. The partial realized gain contains the mismatch, directivity, efficiency, and polarization properties of the antenna. The antenna identity expresses how the performance depends on the electrical size and shape of the antenna structure. It is also the starting point for several antenna bounds. In this paper, the identity, its associated physical bounds, and computational aspects of the polarizability dyadics are discussed.

Sum rules are useful in many branches of physics and engineering as they relate all spectrum parameter values with their asymptotic expansions. Properties of the dynamic response can hence be inferred by the, in many cases much simpler, static response. This has e.g., been used for lossless matching networks, radar absorbers, extinction cross section, partial realized gain of antennas, high-impedance surfaces, transmission cross section, transmission coefficients, and temporal dispersion of metamaterials. Here, several sum rules and their associated physical bounds are reviewed and it is shown that integral identities for Herglotz functions offer a unified approach in deriving them.